Rigid syntomic cohomology

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MR1809626 (2002c:14035)
Besser, Amnon(4-DRHM)
Syntomic regulators and $p$-adic integration. I. Rigid syntomic regulators.
First paragraph of review: For (possibly non-proper) smooth schemes over a $p$-adic field with good reduction, the author constructs a theory of rigid syntomic cohomology, which is a $p$-adic analogue of the Deligne cohomology. This theory is based on P. Berthelot's rigid cohomology [Invent. Math. 128 (1997), no. 2, 329--377; MR1440308 (98j:14023)], in the same way as the Deligne cohomology is based on the de Rham/Hodge cohomology. He also constructs a syntomic regulator from $K$-theory to his rigid syntomic cohomology, which is an analogue of Beilinson's regulator. Probably one may expect to formulate a $p$-adic version of the Beilinson conjecture, which should predict a relation between special values of $p$-adic $L$-functions and his syntomic regulator. The author obtains a result in this direction for CM elliptic curves [Part II, Israel J. Math. 120 (2000), part B, 335--359; MR1809627 (2002c:14036); see the following review].